CS134: Networks Spring Random Variables and Independence. 1.2 Probability Distribution Function (PDF) Number of heads Probability 2 0.

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1 CS134: Networks Spring 2017 Prof. Yaron Singer Section 0 1 Probability 1.1 Random Variables and Independence A real-valued random variable is a variable that can take each of a set of possible values in R with a certain probability. Two random variables, X and Y are said to be independent if P (X = x Y = y) = P (X = x) for all values x of X and y of Y. 1.2 Probability Distribution Function (PDF) The probability distribution of a real-valued random variable X species the probability that X takes each of its possible values. For example, suppose you ip a coin two times. This simple statistical experiment can have four possible outcomes: HH, HT, TH, and TT. Now, let the variable X represent the number of Heads that result from this experiment. X is a random variable that can take the values 0, 1, and 2. The table below, which associates each outcome with its probability, is the probability distribution of the discrete random variable X: Number of heads Probability Cumulative Probability Distributions (CDF) The cumulative probability distribution of a real-valued random variable X species, for each of the possible values x that X can take, the probability that X takes a value equal to or smaller than x. Let us return to the coin ip experiment. If we ip a coin two times, we might ask: What is the probability that the coin ips would result in one or fewer heads? It would be the probability that the coin ip experiment results in zero heads plus the probability that the experiment results in one head. P (X 1) = P (X = 0) + P (X = 1) = =

2 The table below gives the cumulative probability distribution of the discrete random variable X: Number of heads ProbabilityP (X = x) Cumulative ProbabilityP (X < x) Sanity Test: Are PDF's guaranteed to be non-decreasing? What about CDF's? Solution: PDF's are not guaranteed to be non-decreasing (for example, the PDF for any normal distribution is not non-decreasing). In contrast, CDF's are always non-decreasing The Mean of a Discrete Random Variable The mean or expectation of a real-valued random variable X is a weighted average of the possible values that the random variable can take. For example, if the random variable X can take values in a nite 2 set A R, its mean E(X) is dened by E(X) := x AP (x)x Unlike the sample mean of a group of observations, which gives each observation equal weight, the mean of a random variable weights each outcome x according to its probability, P (x). The mean of a random variable provides the long-run average of the variable, or the expected average outcome over many observations. For example, suppose an individual plays a gambling game where it is possible to lose $1, break even, win $3 or win $10 each time she plays. Let Y be the random variable that takes the value y { 1, 0, 3, 5} when the outcome of the gamble is y. The probability distribution of Y is provided by the following table: Calculate the expectation of Y. y P robability Solution: The mean of the random variable Y can be calculated as follows: µ(y ) = ( 1.3) + (0.4) + (3.2) + (5.1) = = To see this, consider that the CDF is a sum (or integral) over an interval of the PDF; specically, for a discrete random variable dened over (a, b), P (X x) = x P (X = x) and for a continuous random variable, P (X x) = i=a x f(x) where f(x) is the PDF of X. PDFs are nonnegative, so x1 < x2 implies P (X x1) P (X x2), since f(x 2) = P (X = x 2) 0; therefore, CDFs are non-decreasing. 2 If A is innite, we replace sums for integrals. 2

3 In the long run, then, the player can expect to win about 80 cents playing this game the odds are in her favor. Properties of Expectation For any two random variables X and Y, E(X + Y ) = E(X) + E(Y ). For two independent random variables X and Y, E(XY ) = E(X)E(Y ). 1.5 The Variance of a Discrete Random Variable The variance of a real-valued random variable X measures the spread, or variability, of the distribution of X. For example, suppose X can take values in a nite 3 set A R. Then the variance Var(X) of X is dened by Var(X) := x A P (x)(x E(X)) 2 For example, in the gambling game above, the variance of the random variable Y may be calculated as follows: Var(Y ) = ( 1 0.8) (0 0.8) (3 0.8) (5 0.8) = ( 1.8) ( 0.8) (2.2) (4.2) = = = Hence, the variance and standard deviation of Y are 3.96 and 3.96 = 1.99 respectively. Unlike the sample variance of a group of observations, which gives each observation equal weight, the variance of a random variable weighs each outcome x according to its probability, P (x). The standard deviation of the random variable X is the square root of its variance and is sometimes denoted by σ. Note that standard deviation has a more natural interpretation than variance. This is because variance is measured in squared units, whereas standard deviation is in the same units as those in which the random variable itself is measured. Properties of Variance For independent random variables X and Y : Var(X) 0 Var(X) = E(X 2 ) E(X) 2 Var(X + Y ) = Var(X Y ) = Var(X) + Var(Y ) 3 If A is innite, we replace sums for integrals. 3

4 2 Examples of Common Probability Distributions 2.1 Uniform Probability Distribution The simplest probability distribution occurs when all of the values of a random variable occur with equal probability. This probability distribution is called the uniform distribution. For example, suppose that a fair die is tossed. Let X be the random variable that takes the value x when the die lands on x, for x {1, 2, 3, 4, 5, 6}. Since the probability that a fair die lands on each the possible numbers is the same, the random variable X has a uniform distribution. 2.2 Binomial Distribution A Bernoulli trial is a random experiment with exactly two possible outcomes, "success" and "failure", in which the probability of success p is the same every time the experiment is conducted. Closely related to a Bernoulli trial is a binomial experiment, which consists of a xed number of statistically independent Bernoulli trials, each with a probability of success p, and counts the number of successes. A random variable corresponding to a binomial experiment with n Bernoulli trials is denoted by B(n, p), and is said to have a binomial distribution. The probability of exactly k successes in the binomial experiment is given by: ( ) n P (X = k) = p k (1 p) n k k where ( ) n k := n! (n k)!k!, which reads n choose k, is the number of dierent sets of k elements we can choose out of a set of n elements. For example, suppose we create edges between each pair of a set nodes of nodes {a, b, c, d} with probability p. The random variable number of edges in the network is distributed according to a binomial distribution B(n, p) Poisson Distribution The random variable X has a Poisson distribution with parameter λ > 0 if, for k = 0, 1, 2,... : P (X = k) = λk e λ The positive real number λ is equal to the mean and variance of the random variable X. That is, µ(x) = λ and Var(X) = λ. The Poisson distribution can be derived as a limiting case of the binomial distribution as the number of trials goes to innity and the expected number of successes remains xed. Therefore it can be used as an approximation to the binomial distribution if n is suciently large and p is suciently small. There is a rule of thumb stating that the Poisson distribution with λ = np is a good approximation of the binomial distribution B(n, p) if n is at least 20 and p is smaller than or equal to 0.05, and an excellent approximation if n is at least 100 and np is smaller than 10. k! 4

5 Figure 1: graph A a c e b d f a Figure 2: Graphgraph B c e g i k b d f h j l 3 The Exponential and Logarithm Functions The exponential function e x can be dened in various ways that provide useful formulas. Fixing x R, ( lim 1 + x ) n = e x n n The identity above will frequently prove useful in the analysis of networks. When you see an expression like ( 1 + n) x n, remember that you can approximate this expression by e x when n is large. Another denition of e x is given by e x = k=0 The logarithm function log(x) is the inverse of the exponential function e x. That is, for all x we have log(e x ) = x and for all x > 0 we have e log(x) = x. x k k! 4 Graph Theory 4.1 Graph representations. (Jackson and 2.1.2) A graph G = (V, E) is a set of nodes V and a set of node pairs E, called edges. An undirected graph is a graph in which the existence of an edge from a node u to a node v implies the existence of an edge from v to u. A directed graph is a graph in which the existence of an edge from u to v does not necessarily imply the existence of an edge from v to u. Unless stated otherwise, assume the graph is undirected. There are three ways to represent a graph: Picture: A graph can be represented via a picture. For example, Figures 1 and 2 represent 5

6 two dierent graphs 4. Adjacency matrix: A graph can be represented by listing its nodes {1, 2,..., n} and a realvalued n n matrix g, where g ij equals 1 if there is an edge between i and j and equals 0 otherwise 5. For example, graph A in Figure 1 can be represented by listing its nodes {a, b, c, d, e, f} and the adjacency matrix: List of nodes and edges: A graph can be represented by listing its nodes and edges. example, graph A can be represented as a list: 6 For ({a, b, c, d, e, f}, {ac, bd, de, df, ef}) Exercises a. Represent graph B in Figure 2 via its adjacency matrix and via the list of its nodes and edges. b. Graph C has nodes {1, 2, 3, 4, 5, 6} and the adjacency matrix: Represent graph C via a picture and via the list of its nodes and edges 7. c. The representation of graph D in terms of a list of its nodes and edges is: ({a, b, c, d}, {ab, ac, ad, bc, bd, cd}) Represent graph D via a picture and via its adjacency matrix. Solution: 4 Figures are located in the last page of this document. 5 A weighted graph is one whose adjacency matrix contains numbers other than 0 and 1, but for now we will focus on unweighted graphs. 6 Convention: for any objects x1, x 2,..., x n, we denote by {x 1, x 2,..., x n} the set that contains the elements x 1, x 2,..., x n and we denote by (x 1, x 2,..., x n) the ordered set (or list) that contains x 1 as its rst element, x 2 as its second element, and x i as its ith element. The distinction is important. For example, we have {x 1, x 2} = {x 2, x 1}, but in general (x 1, x 2) (x 2, x 1). 7 This type of graph is sometimes called star graph. 6

7 3 4 5 Figure 3: Network C c d Figure 4: Network D a b a. Adjacency matrix: List of nodes and edges: ({a, b, c, d, e, f, g, h, i, j, k, l}, {ab, ac, ad, bc, bc, cd, ef, eg, fh, gh, hj, ik, il, jl, lk}) b. Picture: See Figure 3. List of nodes and edges: c. Picture: See Figure 4. Adjacency matrix: ({1, 2, 3, 4, 5, 6}, {12, 13, 14, 15, 16}) Subgraphs. A subgraph of a given graph G is a graph that can be obtained from G by deleting some of its nodes and edges. For example, the graph ({a, c}, {ac}) is a subgraph of graph A, but the graph 7

8 ({a, b, c}, {bc}) is not. Represent a subgraph of graph B via a picture, via its adjacency matrix, and via the list of its nodes and edges. 4.3 Paths and Cycles (Jackson 2.1.3) A walk in a graph G between nodes i and j is a sequence of edges in G, say (i 1 i 2, i 2 i 3,..., i k 1 i k ), with i 1 = i and i k = j. A path in a graph G between nodes i and j is a sequence of edges in G, say (i 1 i 2, i 2 i 3,..., i k 1 i k ), with i 1 = i and i k = j, and such that each node in the sequence (i 1, i 2,..., i k ) is distinct. A cycle is a walk that starts and ends at the same node, with no node appearing more than once except the starting node, which also appears as the ending node. For example, (uv, vw, wu) is a cycle in graph A. A geodesic between two nodes is a shortest path between them; that is, a path with no more edges than any other path between these nodes. For example, in graph A, (df) is a geodesic between nodes d and f, but (de, ef) is not. 4.4 Components and Connected Subgraphs. (Jackson 2.1.5) A graph is connected if every two nodes are connected by some path. For example, graph A is not connected, but its subgraph ({b, d, e, f}, {bd, df, de, ef}) is connected. A graph is completely connected if it is has an edge between every two nodes. A component C of graph G is a (nonempty) subgraph that (i) is connected and that (ii) is maximal i.e. it is such that all other connected subgraphs of G with some node in common with C are subgraphs of C. 8 For example, both ({a, c}, {ac}) and ({b, d, e, f}, {bd, de, df, ef}) are components of graph A, but ({d, e, f}, {de, df, ef}) is not, since it is a subgraph of the component ({b, d, e, f}, {bd, df, de, ef}) and hence not maximal. a. Write down all the components of graph B via a list of their nodes and edges. Solution: a. and ({a, b, c, d}, {ab, ac, ad, bc, bd, cd}) ({e, f, g, h, i, j, k, l}, {ef, eg, fh, gh, hj, ik, il, jl, kl}) 8 Intuitively, a component is a piece of the graph not connected to anything else. 8

9 4.5 Neighborhood. (Jackson 2.1.7) The neighborhood of a node i is the set of nodes that i is linked to. For example, the neighborhood of the node f in the graph A is {d, e}. Given a set of nodes S, the neighborhood of S is the union of the neighborhoods of all of its members that is, the set of nodes that have at least one edge to some node in S. For example, the neighborhood of the nodes {a, b} in graph A is {c, d}. a. What is the neighborhood of node d in graph B? b. What is the neighborhood of the set of nodes {e, g} in graph B? Solution: a. {a, b, c} b. {e, g, f, h}. Note that {e, g} is a subset of the neighborhood of {e, g} since e neighbors g and vice versa. 4.6 Degree and Graph Density. (Jackson 2.1.8) The degree of a node is the number of links that involve that node. For example, the degree of the nodes a and d in graph A are 1 and 3 respectively. In graph B: a. What is the degree of node a? b. What is the average degree of all the nodes in the graph? c. What is the average degree of node a's neighbors? Solution: a. 3 b. In alphabetical order, each nodes degree is 3,3,3,3,2,2,2,3,2,2,2,3. The average degree is then 2.5. c. The average degree of a's neighbors is 3. 5 Big-O Notation 5.1 Formal Denition Big-O notation is mathematical notation used to describe the behavior of a function as their arguments approach innity. Formally, O( ) is dened as follows: f(x) = O(g(x)) as x if there 9

10 exists a positive constant M such that for all x x, f(x) M g(x). Using this denition, we can see that two linear functions, e.g. f(x) = 5x and g(x) = x, follow the relationship f(x) = O(x) because for M = 5 the denition holds. In fact, for any monomial f(x) = ax b (x 7, 6x, 9, etc.), f(x) = O(x b ). It is useful to think of O( ) as an inequality, i.e. g(x) upper bounds f(x): f(x) " " g(x) in the limit. Then we denote " " with Ω: f(x) = Ω(g(x)) if there exists a positive constant M such that for all x x, f(x) M g(x). For strict inequalities, We say that f(x) = o(g(x)) (small o) if f(x) < M g(x) for x x and similarly for ω. Finally, we say that f(x) = Θ(g(x)) if f(x) = O(g(x)) and f(x) = Ω(g(x)), which can be thought of as "equality". 5.2 In Practice In practice, it is cumbersome to deal with the denition of big-o notation, so we employ a number of rules to simplify our analysis: if f(x) is a sum of terms, we only need to consider the term with the largest growth rate. For polynomials, this means only considering the term with the highest exponent. if f(x) is a product of terms, we can ignore multiplicative constants 5.3 The Growth of Some One-Variable Functions Given two increasing functions, f, g : R R we are interested in answering the question of which of the two functions is larger for large x. Here we will provide some illustrations using the functions x (yellow), x + 5 (magenta), x 5 (cyan), x 5 (red), e x (green), log(x) (blue) and x (black). 10

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13 Figure 9: 13